The instructions in How do I set PATH variables for all users on a server? work to set the PATH for all 'normal' users. However, if I do sudo -s and then printenv PATH the updated path is not shown. I've taken a look at for example Setting TeX Live path for root, but this does not seem to make sense to me, perhaps as I'm from a Windows background. Is there any way to set the truly system-wide path, such that the entries are inherited by absolutely every process running on the system?

Thus whatever is set in the /etc/environment for the path is overridden by sudo.

The manual page for sudoers states:

env_reset If set, sudo will reset the environment to only contain
the LOGNAME, MAIL, SHELL, USER, USERNAME and the SUDO_*
variables. Any variables in the caller's environment
that match the env_keep and env_check lists are then
added. The default contents of the env_keep and
env_check lists are displayed when sudo is run by root
with the -V option. If the secure_path option is set,
its value will be used for the PATH environment
variable. This flag is on by default.

As a workaround, you can use sudo su that will provide a shell with root privileges but containing the right PATH.

Environment variable settings that affect the system as a whole
(rather then just a particular user) should not be placed in any of
the many system-level scripts that get executed when the system or the
desktop session are loaded, but into

/etc/environment - This file is specifically meant for system-wide environment variable settings. It is not a script file, but rather
consists of assignment expressions, one per line. Specifically, this
file stores the system-wide locale and path settings.

Not recommended:

/etc/profile - This file gets executed whenever a bash login shell is entered (e.g. when logging in from the console or over ssh), as
well as by the DisplayManager when the desktop session loads. This is
probably the file you will get referred to when asking veteran UNIX
system administrators about environment variables. In Ubuntu, however,
this file does little more then invoke the /etc/bash.bashrc file.

/etc/bash.bashrc - This is the system-wide version of the ~/.bashrc file. Ubuntu is configured by default to execute this file
whenever a user enters a shell or the desktop environment.